Origins of phase-field crack widening in dynamic fragmentation explained

TL;DR

This work analyzes dynamic fracture within phase-field models, identifying unphysical widening of damage bands caused by trapping of elastic waves in diffusive interfaces. It evaluates a mass-degradation (mass erosion) approach that preserves wave speed in damaged zones and demonstrates that this leads to more interpretable fragmentation patterns and convergence to Griffith-type fracture energy predictions as the regularization length $l_0$ decreases with adequate mesh resolution. The study provides detailed 1D and 2D analyses, showing that wave–damage interactions are central to damage spread, and that mass-preservation can mitigate spurious diffusion and enable sharper crack paths and branching. While promising, the approach sacrifices mass conservation and conflicts with tension–compression energy splits, suggesting future work to balance wave-speed preservation with physical conservation and alternative mechanisms such as plasticity-based formulations.

Abstract

We investigate dynamic crack propagation and fragmentation with the phase-field fracture approach. The method was chosen for its ability to yield crack paths that are independent of the underlying mesh, thanks to the damage regularization zone. In dynamics, we observe a progressive widening of this regularization zone and attribute it to an unphysical trapping of elastic waves. We show that the damage zones do not represent free boundaries accurately and that wave interactions induce additional damage. We reveal how mass erosion, by conserving the elastic wave speed in the damaged regions, can be used to efficiently reduce the spurious diffusion of damage. Furthermore, we provide numerical evidence that dynamically propagating cracks in the phase-field formulation, both with and without mass erosion, converge to the predictions of linear elastic fracture mechanics. For vanishing regularization length, the crack speed and energy release rate become independent of the phase-field regularization length, provided that this length scale is small enough and the mesh fine enough to resolve the process zone.

Figures (22)

• Figure 1: \ref{['fig:phasefield_model']} Regularization of a sharp crack $\Gamma$ by a diffusive damage field $d(\mathbf{x})$ in a continuous domain $\Omega$ subject to traction $\mathbf{f}$ and imposed displacement. The regularized fracture corresponds to the region $\Gamma_{l_0} = \{\mathbf{x} \in \Omega | d(\mathbf{x}) > 0\}$ of width related to $l_0$. \ref{['fig:phasefield_profile']} Optimal damage profile for a 1D crack located at $x=0$. Note the limited support of the damage for the AT1 formulation.
• Figure 2: Geometry and boundary conditions for the fragmentation of an expanding spherical membrane, and the equivalent 2d problem, a plate in radial expansion.
• Figure 3: \ref{['fig:ex_frag_full_0']}-\ref{['fig:ex_frag_full_3']} Damage field without mass degradation at four different time steps. \ref{['fig:ex_frag_deg_0']}-\ref{['fig:ex_frag_deg_3']} Damage field with mass degradation at four different time steps.
• Figure 4: Geometry and boundary condition of the 1D bar test. An impulse is sent from the left toward a free boundary (right), with or without damage.
• Figure 5: \ref{['fig:strain_reflect_ref_large']} c-t diagram of the reference case without damage and \ref{['fig:strain_reflect_ref_zoom']} close up on the highlighted rectangular area. We observe a sharp reflection of the wave at the free boundary with a change of sign.